Show the series $\sum\limits_{n=1}^\infty\frac{1}{2+\sqrt{n}}$ diverges

Question

I know intuitively why this series diverges but I can't really get a proof.

So I am trying to use that the fact that: $\sum\limits_{n=1}^\infty\frac{1}{2+\sqrt{n}} > \sum\limits_{n=1}^\infty\frac{1}{2+{n}} $.

And then from there I want to get it down to something like 1/n, which is a p-series with p=1 therefore it converges, and then use the comparison test to show the original sequence diverges. However I am stuck finding in this middle bit.

Any help would be appreciated.

Answer 1

You already have the answer. $\sum_{n=1}^\infty\frac1{2+n}$ is a tail of the divergent harmonic series and thus diverges itself. The original series therefore diverges by the limit comparison test.